Friday, March 24, 2006

There is much talk these days about Israel's particular method of allocating mandates based on election returns. The relevant law is known as Bader-Ofer (after the two MKs who proposed it). So here comes a little tutorial on the much misunderstood Bader-Ofer Law.

First of all, Bader and Ofer didn't invent the system; they simply proposed the adoption of a vote allocation system known as the Hagenbach-Bischoff method, which itself is a variation on the D'Hondt method (both named after 19th century physicists).

How does the method work? If I could easily insert equations into this blog, my life would be easier. I'll do my best.

Step 1: Let's call the total number of votes cast T. Any party that doesn't get at least T/50 votes is eliminated. (50 is an adjustable parameter; last time it was 66.666.) Let T' be the number of votes left after all the votes for eliminated parties are discarded.

Step 2: Let M = T'/120. This is the nominal cost of a single mandate. Let Xn be the number of votes received by party n. Then party n should in theory get Xn/M mandates. The problem is that Xn/M is almost certainly not a whole number but rather something like 13.356. So the first thing we do is give party n, the integer part ("floor") of Xn/M. In our example, that would be 13.

Step 3: Well, since the fractional mandates that each party theoretically got remain unallocated, the total number of mandates given out in Step 2 will be somewhat less than 120. In fact, since the fractional remainders should distribute normally around 1/2, the total number of unallocated seats should be equal to approximately 1/2 the number of parties that cross the threshold in Step 1. (This year that means around 5 mandates.) Now you might think that these should be distributed according to the size of the fractional remainder that each party got. But no! Rather, we use a recursive procedure in which the unallocated mandates are allocated one at a time in the following way:
Let Yn be the number of mandates allocated to party n so far. Initially, Yn is simply the integer part of Xn/M. (By the way, here is the difference between D'Hondt and Hagenbach-Bischoff; in D'Hondt, Yn is initialized to 0 rather than Xn/M.) Now (drum roll...here comes the main point) the next mandate is allocated to the party for which Xn/(Yn+1) is maximal.
The idea is that Xn/Yn is the average cost paid by party n for a mandate and this method keeps the range of such costs over the different parties as narrow as possible.
For example, suppose the total number of votes is 120,000 and party 1 gets 32,340 votes and party 2 gets 14,550 votes (for simplicity, let's ignore the other parties). The nominal cost of a mandate is 1000 so initially party 1 gets 32 mandates and party 2 gets 14 mandates. Now, we want to allocate the mandates lost to the fractional remainders. So we compare 32,340/33 (=980) to 14,550/15 (=970). The result is that party 1 gets the next mandate even though party 2 had a higher remainder.

(Note that the denominators in the above fractions are always 1 more than the current allocation for that party; the point is to determine the hypothetical cost of a mandate after a party were to get it. Another way to think about this is to let Qn be the number of votes that party n is missing to reach the next mandate. Instead of giving the mandate to the party for which Qn is minimal, we give it to the party for which Qn/(Yn+1) is minimal.)

Overall, as is evident in our example, this favors larger parties to a considerable extent (the larger Yn the better your chances of having the minimal Qn/(Yn+1)). For that reason, two parties can agree to be treated as a single party for the purpose of the above calculation. The entire procedure remains otherwise identical. Once, the combined party has been allocated its additional mandates, these are divided between the two consitutent parties using the same method. So when a party makes such an agreement, they should ideally do so with a like-minded party that is smaller than them but still big enough to cross the threshold in Step 1.

The main part of all this that should impact on how you vote is Step 1. If you vote for a party that doesn't make the cut in Step 1, the total number of votes that count (T') is diminished and therefore so is the nominal cost of a mandate (M). So all the other parties increase their mandates proportionally. The increase for larger parties is proportionally greater than for smaller ones. All the rest is just details.

3 Comments:

Anonymous Moshe S said...

Thank you for the explanation. It's nice to know that there are some reasonable things about politics here.

I have been hoping for a post with your thoughts on the parties. In my town the religious will distribute their votes between two parties, neither of which anyone can get excited about. I am sorry to cast my fist vote as an Israeli against rather than for.

Many of my neighbors feel strongly that we should vote on principles, even if the "principled" party has yet to express any serious, realistic plan.

Is this election bound to be a loser, with the best possible outcome a government that can't accomplish much for the next four years?

12:18 PM  
Blogger bar_kochba132 said...

Moshe, I believe the best outcome we can hope for , is as you say, a weak paralyzed gov't. In this post-Zionist era, strong governments bulldoze Jewish communities and take on "sacred cows" like the religious status quo. One interesting outcome is the expected disappearance of Shinui (which split into 2 antagonistic factions) and the fact that MERETZ is also not expected to do well. THIS IS BAD FOR THE RELIGIOUS PARTIES! Sounds strange, doesn't it? However, the religious parties have traditionally used the existence of these anti-religious groups to justify selling out their voters in return for cash and cabinet seats.....Yahdut HaTorah (United Torah Judaism) justified voting to destroy Gush Katif by saying, "if we don't go with Sharon, then SHINUI will, and then there will be civil marriage, etc", which of course, isn't true.

In any event, I have pretty much decided to vote for Marzel's party, SIMPLY AS A PROTEST VOTE...first of all, I don't agree with his Kahanist positions and I know very well that if he were to get into the Knesset, he would be neutralized and put under herem by THE RIGHT which will say he is an extremist.
I justify voting for him ONLY because I am so disgusted with the Ihud Leumi/MAFDAL conglomeration (I vote Ihud Leumi the last 2 times) which was totally ineffective and refused to put much new blood in their list, plus I can't support the Likud, since it has Trojan Horses like Steve Shalom who is acting as an agent for Olmert in order to destroy the Likud from within, and since it also still has the same Central Committee that supported Sharon's destruction of Gush Katif. A pretty bleak situation.

1:56 PM  
Anonymous Moshe S said...

bar_kochba,

I have heard your sentiments about IL/Mafdal many times. People are voting for them again anyway, hoping that somehow this time around they'll act more responsibly.

I don't understand your decision to throw away your vote. If a weak government is the best we can hope for, then that is what we should vote for. Don't throw away your vote to make a useless gesture of frustration.

6:54 PM  

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